Selecting a survey setting for characterizing a target structure

ABSTRACT

Complex-valued sensitivity data structures corresponding to respective candidate survey settings are provided, where the sensitivity data structures relate measurement data associated with a target structure to at least one parameter of a model of the target structure. Based on the sensitivity data structures, a subset of the candidate survey settings is selected according to a criterion for enhancing resolution in characterizing the target structure.

CROSS REFERENCE TO RELATED APPLICATION

This application claims benefit under 35 U.S.C. §119(e) of U.S.Provisional Patent Application Ser. No. 61/468,781 filed Mar. 29, 2011,which is hereby incorporated by reference.

BACKGROUND

Survey operations can be performed to acquire measurement data forcharacterizing content of a subterranean structure. Examples of surveyoperations include seismic survey operations and electromagnetic surveyoperations. Survey operations can be performed using survey equipmentdeployed at or above an earth surface that is above the subterraneanstructure of interest. Alternatively, at least part of the surveyequipment can be deployed in wellbore(s).

Once survey data is acquired, the survey data can be analyzed tocharacterize the subterranean structure, such as by developing an imageor other representation of the subterranean structure, where the imageor other representation contains properties of the subterraneanstructure.

SUMMARY

In general, according to some embodiments, complex-valued sensitivitydata structures corresponding to respective candidate survey settingsare provided, where the sensitivity data structures relate measurementdata associated with a target structure to at least one parameter of amodel of the target structure. Based on the sensitivity data structures,a subset of the candidate survey settings is selected according to acriterion for enhancing resolution in characterizing the targetstructure.

In general, according to further embodiments, an article comprising atleast one machine-readable storage medium stores instructions that uponexecution cause a system to calculate complex-valued sensitivity datastructures corresponding to respective candidate survey settings, wherethe sensitivity data structures relate measurement data associated witha target structure to at least one parameter of a model of the targetstructure. Based on the sensitivity data structures, a subset of thecandidate survey settings is selected according to a criterion forenhancing resolution in characterizing the target structure.

In general, according to further embodiments, a system includes at leastone storage medium to store complex-valued sensitivity data structurescorresponding to respective candidate survey settings, where thesensitivity data structures relate measurement data associated with atarget structure to at least one parameter of a model of the targetstructure. At least one processor is to select, based on the sensitivitydata structures, a subset of the candidate survey settings according toa criterion for enhancing resolution in characterizing the targetstructure.

In further or other implementations, the criterion is based onidentifying at least one of the candidate survey settings that reducesposterior uncertainty.

In further or other implementations, an iterative procedure isperformed, the iterative procedure including: adding one of thesensitivity data structures to a collection of sensitivity datastructures previously considered; and based on a present content of thecollection of the sensitivity data structures, identifying, according tothe criterion, one of the candidate survey settings corresponding to thepresent content of the collection.

In further or other implementations, the iterative procedure furtherincludes: determining whether a convergence condition has beensatisfied; and if the convergence condition has not been satisfied,continuing with the iterative procedure by adding a further sensitivitydata structure to the collection, and based on a further present contentof the collection, identifying, according to the criterion, one of thecandidate survey settings corresponding to the further present contentof the collection.

In further or other implementations, survey equipment is configuredaccording to the at least one survey setting to perform a surveyoperation of the target structure.

In further or other implementations, a particular one of the candidatesurvey settings includes at least one item selected from the groupconsisting of: type of survey equipment, a position of at least onecomponent of the survey equipment, at least one operationalcharacteristic of the survey equipment.

In further or other implementations, the criterion is according to aratio between a determinant of a prior covariance matrix and adeterminant of a posterior covariance matrix.

In further or other implementations, the sensitivity data structuresinclude information relating to the at least one model parameter that issensitive to a subregion less than an entirety of the target structure.

In further or other implementations, the at least one parameter of themodel accounts for an anisotropic radiation pattern of a scatterer inthe target structure.

In further or other implementations, the at least one parameter of themodel is based on a function that encodes a shape of the scatterer.

In further or other implementations, selecting the subset of thecandidate survey settings comprises selecting a particular candidatesurvey setting plural times to allow for stacking of measurement dataacquired by the particular candidate survey setting.

In further or other implementations, a process including the calculatingand selecting is used to identify, for further processing, a subset ofmeasurement data acquired by a survey arrangement.

In further or other implementations, the sensitivity data structuresrelate perturbations of measurement data to perturbations of the atleast one model parameter.

In further or other implementations, the target structure comprises atarget subterranean structure.

In further or other implementations, the criterion is based on priorcovariance information relating to the model.

In further or other implementations, the criterion is based on a valuederived from a determinant of the prior covariance information and adeterminant of posterior covariance information.

In further or other implementations, the selecting is performed using aniterative procedure in which individual ones of the sensitivity datastructures are added to a collection of sensitivity data structures forconsideration with successive iterations.

Other or alternative features will become apparent from the followingdescription, from the drawings, and from the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

Some embodiments are described with respect to the following figures:

FIGS. 1 and 2 are schematic diagrams of example arrangements forperforming survey operations, according to some implementations;

FIGS. 3 and 4 are flow diagrams of processes of determining one or moresurvey designs, according to various implementations; and

FIG. 5 is a block diagram of an example system that incorporates someimplementations.

DETAILED DESCRIPTION Introduction

To characterize a subterranean structure, it is desirable to collect asmany survey observations (measurement data) as possible to betterunderstand properties of the subterranean structure that is underinvestigation. An observation can include measurement data of a singletrace or a shot gather. A trace refers to measurement data acquired by aseismic sensor (or array of seismic sensors) for a given single shot(activation) of a seismic source. A seismic gather is a collection ofseismic traces that share an acquisition parameter, such as a commonmidpoint or other parameter. Alternatively, an observation can bemeasurement data from a coil, which can include measurement data ofseismic sensor(s) due to multiple shots (multiple activations of aseismic sensor) in some predefined pattern.

In practice, the number of survey observations that is available tooperators is often not optimal for purposes of achieving relatively goodsignal quality, good resolution and adequate spatial coverage of atarget subterranean structure. To enhance accuracy of characterizationof target subterranean structures, pre-survey design can be performed todetermine what settings associated with survey equipment would produceresults of enhanced accuracy. Traditionally, such pre-survey design ismanual in nature and can be labor-intensive and time-consuming.

Pre-survey designs can be based on use of models of subterraneanstructures. However, there can be uncertainties associated with models,due to poor knowledge about the target subterranean structure.Consequently, it is often the case that pre-survey designs are unable todetermine survey equipment settings that provide superior or optimalresults. Examples of survey equipment settings include positions ofsource-receive pairs that are deemed most valuable, or settings relatingto the optimal (or enhanced) frequency bandwidth in the presence ofsignal attenuation, or settings relating to the type of components touse in survey equipment, and so forth. More generally, a “surveysetting” refers to one or more of the following: the type of surveyequipment used, positions of components of the survey equipment,operational settings of the survey equipment, and so forth.

A “survey setting” can also be considered a survey design or a design ofa survey experiment, where a survey design or design of a surveyexperiment refers to an arrangement of survey equipment used to performa survey operation.

Survey observations can be collected from performing seismic surveyoperations. A seismic survey operation involves using at least oneseismic source to generate seismic waves that are propagated into atarget subterranean structure. Reflected waves from the subterraneanstructure are then detected by seismic survey receivers, which can bedeployed at a land surface above the subterranean structure, a seafloorabove the subterranean structure, or towed through a body of water.Seismic survey operations can include land-based seismic surveyoperations, marine seismic survey operations, wellbore seismic surveyoperations, and so forth.

Although reference is made to seismic surveying or acquisitions in thepresent discussion, it is noted that techniques or mechanisms accordingto some embodiments are applicable to other types of survey operations,such as electromagnetic (EM) survey operations. Also, techniques ormechanisms according to some embodiments are applicable to other typesof imaging contexts, such as medical imaging (of human tissue, forexample), imaging of machinery or mechanical structures, and so forth.

In accordance with some embodiments, techniques or mechanisms areprovided for determining survey settings that can provide for enhancedresolution in characterizing a target subterranean structure (or moregenerally, any other type of target structure, including human tissue,machinery, mechanical structure, etc.). The techniques or mechanisms areable to accommodate complex-valued sensitivity matrices, as well ascomplex-valued covariance matrices.

A “sensitivity matrix” refers to a matrix containing parameters thatrelate measurement data (acquired by survey equipment for a subterraneanstructure) to model parameters (of a model that represents thesubterranean structure). A “covariance matrix” contains information thatdescribes the uncertainty regarding a model. Each of the sensitivitymatrices and covariance matrices can be complex-valued, which means thatthe respective matrix contains both real and imaginary values. Asexplained further below, the ability to employ a complex-valuedsensitivity matrix (and in some cases a complex-valued covariancematrix) allows for greater flexibility in characterizing a targetsubterranean structure.

Although the ensuing discussion refers to sensitivity matrices andcovariance matrices, note that in alternative implementations,complex-valued sensitivity data structures and complex-valued covariancedata structures can be used, where a “data structure” refers to anyrepresentation that contains information in a predefined format.

In accordance with some embodiments, a link is established between asensitivity matrix and a survey setting for minimizing (or otherwisereducing) posterior model uncertainties and to maximize (or otherwiseenhance) model resolution. “Posterior” model uncertainties refer touncertainties in a model of a subterranean structure that has beencomputed or adjusted based on observations acquired by a surveyoperation.

Survey Environments

Various different types of seismic survey operations can be performed,including a vertical seismic profile (VSP) survey operation or a surfaceseismic survey operation. A VSP survey arrangement is depicted inFIG. 1. A tool 100, which can include a number of seismic receivers 102(e.g., geophones, accelerometers, etc.) can be positioned in a wellbore104 on a carrier structure 106 (e.g., cable) that is connected tosurface equipment 108. One or more seismic sources 110 (e.g., airguns,vibrators, etc.) can be positioned at the earth surface some distancefrom the wellbore 104. Alternatively or additionally, seismic sourcescan also be provided on the tool 100. When the seismic source(s) 110 is(are) activated, seismic waves travel into a subterranean structure 112that surrounds the wellbore 104. A portion of the seismic waves can bereflected as a result of changes in acoustic impedance in thesubterranean structure 112 due to the presence of various boundaries inthe subterranean structure. The seismic waves that reach the wellbore104 can be detected by the seismic receivers 102 in the wellbore 104,where the signals are recorded by the seismic receivers 102 for laterprocessing in characterizing the subterranean structure 112.

A surface seismic survey arrangement is shown in FIG. 2, which is anarrangement for performing a marine surface seismic survey. A marinevessel 200 can tow a streamer 202 (or multiple streamers) that carryseismic receivers 203. The marine vessel 200 (or a different marinevessel) can tow at least one seismic source 204 that when activatedcauses seismic waves to be propagated into a subterranean structure 206.Reflected waves from the subterranean structure 206 are detected by theseismic receivers 203 in the streamer(s) 202.

Another type of surface seismic survey arrangement is a land-basedsurface seismic survey arrangement, where seismic source(s) and seismicreceivers are deployed on a land surface above the target subterraneanstructure.

According to another example, a seismic survey arrangement includesseismic-guided drilling equipment. The seismic-guided drilling equipmentincludes seismic source(s) and seismic receivers, and informationacquired by the seismic receivers can be used in guiding drilling of awellbore.

Although examples are provided of various different survey arrangements,it is noted that in alternative examples, other survey arrangements canbe used. For example, such other survey arrangements can includeelectromagnetic survey arrangements, or survey arrangements to measuredata of other types of target structures.

Techniques or mechanisms according to some embodiments for determiningsurvey settings that can provide for enhanced resolution incharacterizing a target structure can be applied in the context of anyof the survey arrangements discussed above.

Survey Design Identification Procedures

FIG. 3 is a flow diagram of a process according to some implementations.The process includes calculating (at 302) sensitivity data structurescorresponding to respective candidate survey settings, where each of thesensitivity data structures relates a corresponding set of surveyobservations (of a target subterranean structure) to model parameters(of a model that represents the target subterranean structure). Thecandidate survey settings that correspond to the sensitivity datastructures represent candidate survey designs to be used for acquiringrespective survey observations. Some of the candidate survey settingscan produce superior results (for characterizing the target subterraneanstructure) as compared to other ones of the candidate survey settings.

In some implementations, the sensitivity data structures can be part ofa sensitivity matrix; for example, the sensitivity data structures canbe entries (e.g., rows) of the sensitivity matrix. In someimplementations, the sensitivity data structures can be made up ofdifferent sensitivity matrices. In alternate implementations, otherappropriate data structures can be used in light of the requirements ofthe application, as those with skill in the art will appreciate.

In the ensuing discussion, reference is made to a sensitivity matrix;however, it is noted that in alternative implementations, techniques ormechanisms according to some embodiments can also be applied in thecontext of multiple sensitivity matrices.

The sensitivity data structures can contain complex values (includingboth real and imaginary values). Including complex values in thesensitivity data structures allows for performance of variouscharacterizations that cannot be achieved using sensitivity datastructures that contain just real values. For example, with complexsensitivity data structures, imaging problems formulated in thefrequency domain can be solved.

Based on the sensitivity data structures, the process of FIG. 3 selects(at 304) a subset (less than all) of the candidate survey settingsaccording to a criterion for enhancing resolution in characterizing thesubterranean structure. The selected subset of survey settings caninclude one survey setting or multiple survey settings. The goal is tofind, among the candidate survey settings, the one (or ones) that is(are) likely to provide information about model parameters (of a modelrepresenting a target subterranean structure) that will complement theprior information, which includes a set of already available surveyobservations. In other words, the process of FIG. 3 seeks to identify asurvey design (or survey designs) that is likely to minimize (or reduce)the posterior uncertainty and thus maximize (or enhance) the resolutionof the model parameters (of the model representing the targetsubterranean structure).

The foregoing refers to implementations in which techniques are used toproduce a survey design for achieving a target goal. In furtherimplementations, a survey arrangement may already exist that has beenused to acquire measurement data regarding a subterranean structure.Techniques according to some embodiments, such as according to FIG. 3,can be used to help identify a subset of the measurement data thatcontains more useful information—the notion here is that processing ofthe subset of the measurement data is more efficient than processing thefull measurement data. In this manner, accelerated or more rapidprocessing of measurement data can be achieved, such as for purposes ofquality control, providing a preliminary representation (e.g., image) ofthe subterranean structure for quick review, and so forth. Techniquesaccording to some embodiments, such as according to FIG. 3, can identifya subset of the survey arrangement that satisfies the criterion forenhancing resolution in characterizing the subterranean structure. Thesubset of measurement data produced by the identified subset of thesurvey arrangement is then subject to further processing.

The ensuing discussion describes in detail how a sensitivity matrix Gcan be derived in accordance with an example embodiment. The sensitivitymatrix G is set further forth below in Eq. 5.

Consider a viscoelastic earth model (for representing a targetsubterranean structure) characterized by mass density ρ(x) and complexstiffness coefficients c_(ijkl)(x), where x is a Cartesian positionvector. The mass density and stiffness coefficients are examples ofproperties or parameters of a subterranean structure that can beincluded in a model of the subterranean structure. In other examples, amodel of a subterranean structure can include other properties orparameters.

The variables i, j, k, l represent four respective dimensions, includingthe Cartesian dimensions x,y,z and a time dimension. The medium (of thetarget subterranean structure) is assumed to be contained in asemi-infinite domain of interest V. V represents a volume containing thesubterranean structure, and is bounded by a closed surface

=

∪

, where

is a traction-free surface and

is a surface at infinity where the radiation condition holds. To accountfor attenuation in elastic wave propagation, the stiffness coefficientsc_(ijkl)(x) are considered to have complex values. In addition, thestiffness coefficients are assumed to be time-independent withsymmetries such that c_(ijkl)=c_(jikl)=c_(klij)=c_(ijlk).

Let u=(u₁, u₂, u₃) be the three-component displacement field (where thedisplacement field contains displacement data measured by seismicsensors), with u_(i)(x, ω) being the i^(th) component of displacement atpoint x and frequency ω. In other examples, other types of measurementdata (instead of displacement data) can be measured, such as velocitiesand acceleration. Here, x=(x₁, x₂, x₃)≡(x, y, z)εV. In the frequencydomain, the elastic wave equation with attenuation can be written as

ω²ρ(x)u _(i)(x,ω)+∂_(j) [c _(ijkl)(x)∂_(k) u _(l)(x,ω)]=f_(i)(x,ω)+∂_(j) M _(ij)(x,ω),  (Eq. 1)

where ∂≡∂/∂x_(j), f_(i) is a volume density of force applied in thei^(th) axis direction, M_(ij) is a moment density indexed by the twoaxis directions i and j, and where the Einstein summation convention isused. The solution of the elastic wave equation (Eq. 1) can be writtenas a function of the attenuative outgoing Green function

, (x, x′, ω), as follows:

u _(i)(x,ω)=−∫_(v)

(x,x′,ω)[f _(n)(x′,ω)+∂_(j) M _(nj)(x′,ω)]dV′,  (Eq. 2)

where V′≡V(x′), and x′ represents possible locations of the medium(target subterranean structure). The Green function is a function usedto solve inhomogeneous differential equations subject to initialconditions or boundary conditions. The summation over the indices n andj accounts for volume density forces applied in the three possibledirections (nε{1,2,3}), and for the moment densities with (n, j)ε{(1,2),(1,3), (2,3)}.

Now, let u⁽⁰⁾=(u₁ ⁽⁰⁾, u₂ ⁽⁰⁾, u₃ ⁽⁰⁾) be the displacement field due tosources {F,∇M}≡{f_(i),∂_(j)M_(ij)}, in a medium represented by the modelvector m⁽⁰⁾≡ρ⁽⁰⁾, {c_(ijkl) ⁽⁰⁾}). Consider that a model m (whichrepresents a target subterranean structure) is obtained by adding aperturbation model represented by the vector Δm=(Δρ, {Δc_(ijkl)}) tom⁽⁰⁾ so that

ρ=ρ⁽⁰⁾+Δρ  (Eq. 3)

c _(ijkl) =c _(ijkl) +Δc _(ijkl.)

In Eq. 3, each of the density ρ and stiffness coefficient c_(ijkl) isbased on a previous value of the density and stiffness coefficient,respectively, as modified by a perturbation (change) of the density (Δρ)and stiffness coefficient (Δc_(ijkl)), respectively. In the foregoing,m⁽⁰⁾ can be considered the initial model of the subterranean structure,and m is considered an updated model after an adjustment of m⁽⁰⁾ usingthe model perturbation represented by the vector Δm. Alternatively, m⁽⁰⁾can be considered the previous model (which has been updated in aprevious iteration, where the previous model is to be further updated toupdated model m in the present iteration).

Under the Born approximation, a perturbation of the model parameters,Δm, such that m=m⁽⁰⁾+Δm, leads to a perturbation of the displacementfields (measurement data), Δu≡(Δu₁, Δu₂, Δu₃) so that u=u⁽⁰⁾+Δu.Inserting the expressions of Eq. 3 into Eq. 1, and assuming singlescattering and neglecting higher order terms yields the perturbeddisplacement fields Δu_(i) that are also solutions of the attenuativeelastic wave equation with) m⁽⁰⁾ as the propagation medium and Δm as thescattering source. Thus, the Born approximation perturbationdisplacement fields can be written as

Δu _(i)(x,ω)=ω²∫_(V)

(x,x′,ω)u _(j) ⁽⁰⁾(x′,ω)Δρ(x′)dV′−∫ _(V)∂′_(k)[

(x,x′,ω)]Δc _(nklj)(x′)∂′_(l) u _(j) ⁽⁰⁾(x′,ω)dV′.  (Eq. 4)

where

=(x, x′, ω) is the outgoing Green function in the unperturbed medium,and u_(i) ⁽⁰⁾(x′, ω) is the unperturbed incident wave, expressed asu_(i) ⁽⁰⁾(x, ω)=−∫_(V)Σ_(j)

(x, x′, ω)s_(j)(x′, ω)dV′. Also, Δρ represents contrasts in density, andΔc_(njkl) represents a contrast in a stiffness coefficient.

As noted above, a goal according to some embodiments is to select asurvey design (survey setting) that would yield, a posteriori, the best(or enhanced) resolution of model parameters. Assuming a model containsmass density ρ(x) and stiffness coefficients c_(ijkl)(x), as notedabove, for example, then the goal is to find a survey setting that islikely to be the most sensitive to the model parameters Δρ andΔc_(ijkl). To that end, Eq. 4 set forth above carries two pieces ofinformation. Eq. 4 shows that the Born approximation perturbationdisplacement fields Δu are linearly related to the perturbation modelparameters Δm=(Δρ, {Δc_(ijkl)}) (which, in general, has 1+6×6components, for example). Eq. 4 also expresses the fact thatdisplacement data, recorded at receiver location x, results fromcontributions from wavefields possibly coming from the locations x′ inthe medium.

The linearity between the Born approximation displacement fields and themodel parameters implies that Eq. 4 can be written in a compact form as

Δu=GΔm,  (Eq. 5)

where G, the sensitivity matrix, is a possibly complex N×(1+6×6)Msensitivity matrix. Here, N and (1+6×6)M, for example, are the number ofmeasurement observations and the number of model parameters,respectively. For u_(i)(x_(j), ω), the i^(th) displacement componentdata recorded at a given receiver location x_(j), the elements G_(ij) ⁹²and G_(ij) ^((c)) of the sensitivity matrix G represent thecontributions to be associated with the mass density ρ(x′) and thestiffness coefficients c_(ijkl)(x′), respectively. These elements, whichmake up the sensitivity matrix G, are defined by the functionalderivatives

G ij ( ρ ) ≡ δ  [ Δ   u i  ( x j , ω ) ] δ  [ Δρ  ( x ′ ) ] = ω 2 il ( 0 )  ( x j , x ′ , w )  u l ( 0 )  ( x ′ , w ) ( Eq .  6 ) Gij ( c ) ≡ δ  [ Δ   u i  ( x j , ω ) ] δ  [ Δ   c nklp  ( x ′ )] = - ∂ k ′  [ i   n ( 0 )  ( x j , x ′ , w ) ]  ∂ l ′  u p ( 0 ) ( x ′ , w ) . ( Eq .  7 )

where

represents the outgoing Green function, which can be complex. As notedabove, the Green function is a function used to solve inhomogeneousdifferential equations subject to initial conditions or boundaryconditions.

The sensitivity matrix G thus relates measurement data (acquired for atarget structure) to parameters of a model (that represents the targetstructure). More specifically, according to Eq. 5, the sensitivitymatrix G relates perturbation measurement data (Δu) to a perturbation ofthe model (Δm).

The following describes the link between the sensitivity matrix G andsurvey settings to minimize (or reduce) the posterior modeluncertainties associated with solving Eq. 5, and to maximize (orenhance) the corresponding model resolution.

Assuming that what is known about the prior model m⁽⁰⁾(which can be theinitial model or the previous model) and data uncertainties are theirmeans and covariances, the optimal (or enhanced) approximation for theprior model and data noise distributions is the multivariate normaldistribution. In such circumstances, the Bayesian least-square solutionof the linear system of Eq. 5 is a Gaussian distribution

(

, {tilde over (C)}) with center

=CG ^(H)(GCG ^(H) +C _(D))⁻¹ Δu _(obs)  (Eq. 8)

and covariance

{tilde over (C)}=(G ^(H) C _(D) ⁻¹ G+C ⁻¹)⁻¹,  (Eq. 9)

where the superscript H denotes a Hermitian transpose,

represents a mean solution for the model of the target subterraneanstructure, Δu_(obs)=Δu+e is a vector of noisy measurements with noise,e˜

(0, C_(D)) and the uncertainties on the prior model are described by

(0, C). In Eqs. 8 and 9, C_(D) represents the prior covariance matrix onthe measurement data, which represents uncertainty regarding measurementdata. Also, C represents the prior covariance matrix on the model, whichrepresents uncertainty regarding model parameters. A “prior” covariancematrix describes uncertainty before acquisition of data and selection ofa particular survey setting. The matrix {tilde over (C)} is a posteriorcovariance matrix, which represents uncertainty following acquisition ofdata and selection of a particular survey setting. In the general caseC_(D), C, and {tilde over (C)} are positive-definite, Hermitian matricesof complex Δu (measurement data acquired by survey equipment) and Δm(model parameters). Eq. 8 provides a mean solution for the modelparameters.

The posterior covariance matrix {tilde over (C)} helps to analyze howwell the system has been resolved due to selection of a particularsurvey setting for acquiring survey observations. The diagonal elementsof {tilde over (C)} are the variances (commonly referred to as errorbars) on the posterior values of the model parameters

. The off-diagonal elements are the covariances and provide a measure onhow well a given model parameter has been resolved independently of theothers. Ideally, a perfectly resolved system would lead to a posteriorcovariance matrix {tilde over (C)} with zero error bars on the resolvedmodel parameters. In such cases, the associated determinant, det({tildeover (C)}), of the posterior covariance matrix is zero.

If one then knew the true model, then

=Δm_(exact). However, in practice, the posterior model is more likely tobe a filtered version of the exact model. Such a filtered version of

is represented as

=RΔm _(exact),  (Eq. 10)

wherein R is the resolution matrix:

$\begin{matrix}\begin{matrix}{R \equiv {{{CG}^{H}\left( {{GCC}^{H} + C_{D}} \right)}^{- 1}G}} \\{= {I - {\overset{\sim}{C}{C^{- 1}.}}}}\end{matrix} & \left( {{Eq}.\mspace{14mu} 11} \right)\end{matrix}$

In Eq. 11, I is the identity matrix, where the identify matrix has isthe main diagonal and 0s elsewhere. The resolution matrix R representshow a survey system distorts a model from an exact model of a targetsubterranean structure. The columns of the resolution matrix indicatehow a delta function in the exact model would be mapped in the estimatedmodel. In some examples, the columns include the point-spread functions(PSF) associated with R. The rows of the resolution matrix R indicatehow the exact model parameters are averaged to obtain a given estimatedmodel parameter. The rows are sometimes referred to as theBackus-Gilbert averaging kernels. Also it follows from Eq. 11 that

{tilde over (C)}=(I−R)C,  (Eq. 12)

which shows that any departure of {tilde over (C)} (the posteriorcovariance matrix) from the null matrix indicates imperfect resolution,as represented by R.

For a set of observations, {Δu_(n)} (where n represents a presentiteration and n iterates from 1 to some predefined number), collectedbased on a corresponding survey setting {ξ_(n)}≡{ξ:ξεΞ}, theleast-square solution of Eq. 5 is the Gaussian distribution

(

, {tilde over (C)}) if one assumes that the uncertainties associatedwith the prior model and with the observations are Gaussian anddescribed by

(0, C) and

(0, C_(D)), respectively. In the foregoing, Ξ represents the universe ofcandidate survey settings that are considered.

Techniques or mechanisms according to some embodiments seek to find,among a plurality of candidate survey designs (survey settings), the oneor ones that is (are) likely to provide information about the modelparameters that will better complement the prior information, inparticular, the set of already available observations. The result is theidentification of a setting that may minimize (or reduce) the posterioruncertainty and thus maximize (or enhance) the resolution of the modelparameters of the model that represents the target subterraneanstructure. To achieve the above, the following objective function isused:

$\begin{matrix}{{\xi_{opt}^{(C)} \equiv {\arg \; {\max\limits_{\xi \in \Xi}\frac{\det (C)}{\det \left( \overset{\sim}{C} \right)}}}},{{for}\mspace{14mu} a\mspace{14mu} {fixed}{\Xi }},} & \left( {{Eq}.\mspace{14mu} 13} \right)\end{matrix}$

where |•| stands for the cardinal and det(•) for the determinant, andwhere Eq. 13 seeks to find a survey setting from ξ that results in amaximum of the ratio det(C)/det({tilde over (C)}). A determinant of amatrix (which in Eq. 13 includes C or {tilde over (C)}) is a valuecomputed from the elements of the matrix by predefined rules. The ratioof determinants, det(C)/det({tilde over (C)}), is referred to as the“C-norm.” The objective function of Eq. 13 is equivalent to minimizingtr(R) (trace of R), which effectively reduces the distortion representedby R. Said differently, the objective function constitutes a criterionfor enhancing resolution in characterizing the target structure, byreducing distortion represented by R.

The process of identifying a survey setting (defining a survey design toacquire a respective observation), or multiple survey settings, is aniterative process that is performed on an observation-by-observationbasis. Note that different observations are acquired using surveyequipment having different survey settings. Thus, in the ensuingdiscussion, reference to selecting an observation is the equivalent ofselecting a respective survey setting.

The iterative procedure for identifying a survey setting (or surveysettings) on an observation-by-observation basis is depicted in FIG. 4.An iteration variable, n, is initialized to zero (at 402). This variableis incremented with each successive iteration. Let G_(n) be asensitivity matrix whose rows correspond to the observations chosen sofar from the rows of the sensitivity matrix G, and let g_(n+1) ^(T) be arow of G corresponding to a candidate observation. In the beginning,G_(n) is an empty matrix (when n=0), with no observations. However, asobservations are chosen from G and added to G_(n), G_(n) isincrementally updated one entry (e.g., row) at a time, by adding (at404) the current candidate observation, g_(n+1) ^(T), which is chosenfrom G. Thus, at iteration n+1, the sensitivity matrix G_(n+1) of theexperiment under consideration is given by the block matrix

$\begin{matrix}{{G_{n + 1} = \begin{bmatrix}G_{n} \\g_{n + 1}^{T}\end{bmatrix}},} & \left( {{Eq}.\mspace{14mu} 14} \right)\end{matrix}$

where n=1, 2, . . . . In the most general case, the data priorcovariance matrix, C_(D), may be written as

$\begin{matrix}{\left( C_{D} \right)_{n + 1} = \begin{bmatrix}\left( C_{D} \right)_{n} & c_{n + 1} \\c_{n + 1}^{H} & \sigma_{n + 1}^{2}\end{bmatrix}} & \left( {{Eq}.\mspace{14mu} 15} \right)\end{matrix}$

wherein (C_(D))_(n) is the data prior covariance matrix of the baseexperiment data (the measurement data prior to addition of the currentcandidate observation g_(n+1) ^(T)), σ_(n+1) ² is the variance of thedata measurement that corresponds to the new candidate observation,g_(n+1) ^(T), and c_(n+1) is the vector whose components are thecovariance terms of the new candidate observation, g_(n+1) ^(T).

It can be shown that the objective function in Eq. 13 takes on theiterative form

$\begin{matrix}{{\frac{\det \left( C_{n} \right)}{\det \left( C_{n + 1} \right)} = {\left( {1 + {g_{n + 1}^{T}A^{- 1}h^{*}}} \right){{\det \left( {I + {{BG}_{n}C_{n}G_{n}^{H}}} \right)}\left\lbrack {1 + {{k^{T}\left( {A^{- 1} - \frac{A^{- 1}h^{*}g_{n + 1}^{T}A^{- 1}}{1 + {g_{n + 1}^{T}A^{- 1}h^{*}}}} \right)}g_{n + 1}^{*}}} \right\rbrack}}},\mspace{20mu} {where}} & \left( {{Eq}.\mspace{14mu} 16} \right) \\{\mspace{79mu} {{A \equiv {C_{n}^{- 1} + {G_{n}^{H}{BG}_{n}}}},}} & \left( {{Eq}.\mspace{14mu} 17} \right) \\{\mspace{79mu} {{B \equiv {\sigma_{n + 1}^{- 2}{\varphi_{n + 1}^{- 1}\left( C_{D} \right)}_{n}^{- 1}c_{n + 1}{c_{n + 1}^{H}\left( C_{D} \right)}_{n}^{- 1}}},}} & \left( {{Eq}.\mspace{14mu} 18} \right) \\{\mspace{79mu} {{h^{*} \equiv {\sigma_{n + 1}^{- 2}{\varphi_{n + 1}^{- 1}\left\lbrack {g_{n + 1}^{*} - {{G_{n}^{H}\left( C_{D} \right)}_{n}^{- 1}c_{n + 1}}} \right\rbrack}}},}} & \left( {{Eq}.\mspace{14mu} 19} \right) \\{\mspace{79mu} {{k^{T} \equiv {\sigma_{n + 1}^{- 2}\varphi_{n + 1}^{- 1}{c_{n + 1}^{H}\left( C_{D} \right)}_{n}^{- 1}G_{n}}},}} & \left( {{Eq}.\mspace{14mu} 20} \right) \\{\mspace{79mu} {{\varphi_{n + 1} \equiv {1 - {\sigma_{n + 1}^{- 2}{c_{n + 1}^{H}\left( C_{D} \right)}_{n}^{- 1}c_{n + 1}}}},}} & \left( {{Eq}.\mspace{14mu} 21} \right)\end{matrix}$

where the superscript * stands for the complex conjugate. In Eq. 16,C_(n), represents the prior covariance matrix on the model, whereasC_(n+1) represents the posterior covariance matrix. For uncorrelateddata noise, c=0 and, consequently, Eq. 16 reduces to a simplerexpression

$\begin{matrix}\begin{matrix}{\frac{\det \left( C_{i} \right)}{\det \left( C_{i + 1} \right)} = {1 + {\sigma_{n + 1}^{- 2}g_{n + 1}^{T}C_{n}g_{n + 1}^{*}}}} \\{= {1 + {\sigma_{n + 1}^{- 2}g_{n + 1}^{H}C_{n}^{*}g_{n + 1}}}}\end{matrix} & \begin{matrix}\left( {{Eq}.\mspace{14mu} 22} \right) \\\; \\\left( {{Eq}.\mspace{14mu} 23} \right)\end{matrix}\end{matrix}$

where the Hermiticity of the covariance matrix has been invoked.

From among the observations corresponding to the entries of G_(n), theprocedure of FIG. 4 identifies (at 406) an entry i of G_(n) thatmaximizes the following objective function derived from Eq. 22:

$\begin{matrix}{\frac{\det \left( C_{i} \right)}{\det \left( C_{i + 1} \right)} = {1 + {\sigma_{i + 1}^{- 2}g_{i + 1}^{T}C_{n}{g_{i + 1}^{*}.}}}} & \left( {{Eq}.\mspace{14mu} 24} \right)\end{matrix}$

Next, n is incremented (at 408) by 1 to perform continue to the nextiteration.

The procedure next determines (at 410) if a convergence condition hasbeen satisfied. If not, then the procedure proceeds to perform tasks404, 406, and 408 for the next iteration. If the convergence conditionhas been satisfied, then the procedure stops (at 412), and the surveysetting corresponding to entry i of G_(n) identified at 406 is returnedas the selected optimal (or enhanced) survey setting.

An example convergence condition can include the following condition:adding a new observation to the experiment, as represented by G_(n), nolonger results in significant improvement of the ratio defined by Eq.22; for example, if the improvement of the ratio defined by Eq. 22 isless than some predefined threshold, then that is an indication that theconvergence condition has been met. Another example convergencecondition can be that the number of observations added to G_(n) hasexceeded some predefined maximum number.

A benefit of using the iterative procedure of FIG. 4 is that theentirety of the sensitivity matrix G does not have to be considered tofind the optimal (or enhanced) survey setting(s). The sensitivity matrixG can be large and thus it may be computationally expensive to considerthe entirety of the entries of the sensitivity matrix G. In accordancewith some implementations, by considering the partial matrix G_(n), andstopping when a convergence condition has been met, computationalefficiency can be enhanced.

The foregoing has described Bayesian techniques for selecting surveydesigns that optimally maximizes (or enhances) model parameterresolution for characterizing target structures, such as subterraneanstructures, human tissues, machinery, mechanical structures, and soforth. The techniques according to some implementations are referred toas Bayesian D-optimality-based techniques, based on usage of anobjective function (e.g., Eq. 13 or 22) that seeks to maximize a ratioof determinants of covariance matrices. The techniques are able toselect observation(s) that is (are) likely to reduce the forecastuncertainties on the model parameters.

In practice, an observation can be associated with multiple measurements(e.g., a seismic sensor can record the waveforms of both P-waves andS-waves, where P-waves refer to compression waves, and S-waves refer tosheer waves). In cases where each observation has multiple measurements,the foregoing techniques can augment the survey design with k (k≧2)measurements at a time. Let Γ=[Γ₁ Γ₂ . . . Γ_(k)]^(T) be the matrixwhose rows Γ_(i) ^(T), 1≦i≦k<<M, are the sensitivity kernels of therelevant seismic sensors. For uncorrelated data, i.e., for

$\begin{matrix}{{\left( C_{D} \right)_{n + k} = \begin{bmatrix}\left( C_{D} \right)_{n} & 0 \\0 & S\end{bmatrix}},} & \left( {{Eq}.\mspace{14mu} 25} \right)\end{matrix}$

wherein S=(σ_(i) ²δ_(ij))_(i,j≦k), where σ_(ij), is the Kronecker delta,a one-time rank-k augmentation can be defined as

$\begin{matrix}{\frac{\det \left( C_{n} \right)}{\det \left( C_{n + k} \right)} = {\frac{\det \left( {S + {\Gamma \; C_{n}\Gamma^{H}}} \right)}{\det (S)}.}} & \left( {{Eq}.\mspace{14mu} 26} \right)\end{matrix}$

Marginal Covariance

Techniques or mechanisms according to some embodiments can be modifiedto focus on a particular subregion of interest within a targetsubterranean structure (where the particular subregion is less than theentirety of the target subterranean structure). In other words, ratherthan focus on the entire target subterranean structure, an enhanced (oroptimal) survey design (or survey designs) can be identified for asubregion in the target subterranean structure.

To focus on a particular subregion of interest (rather than the entiretarget subterranean structure), the prior covariance matrix C, whichdescribes quantitative uncertainties in the model parameters, can beused to specify the particular subregion of interest by assigning ahigher prior uncertainty to that particular subregion. A designcriterion can be used to preserve the C matrix description of thequantitative uncertainties while defining the particular subregion ofinterest separately. For this particular subregion of interest, theposterior marginal covariance matrix for the model at iteration n isgiven by T_(n), which is a marginal covariance matrix (marginal in thesense that it focuses on a subregion of a model). The objective functionin this case is to find the design

$\begin{matrix}{{\xi_{T}^{({opt})} \equiv {\arg \; {\max\limits_{\xi \in \Xi}\frac{\det \left( T_{n} \right)}{\det \left( T_{n + 1} \right)}}}},{{for}\mspace{14mu} a\mspace{14mu} {fixed}\mspace{14mu} {\Xi }},} & \left( {{Eq}.\mspace{14mu} 27} \right)\end{matrix}$

instead of ξ_(C) ^(opt)) in Eq. 13 or 22. Note that the objectivefunction of Eq. 27 is based on a ratio of determinants of the T_(n) andT_(n+1) matrices, rather than the by C_(N) and C_(n+1) of the objectivefunction of Eq. 13 or 22 discussed further above. The above ratio ofdeterminants in Eq. 27 is referred to as the T-norm.

With uncorrelated measurement noise the posterior model covariancematrix at iteration n+1 reduces to

$\begin{matrix}{C_{n + 1} = {C_{n} - {\frac{\sigma_{n + 1}^{- 2}C_{n}g_{n + 1}^{*}g_{n + 1}^{T}C_{n}}{1 + {\sigma_{n + 1}^{- 2}g_{n + 1}^{T}C_{n}g_{n + 1}^{*}}}.}}} & \left( {{Eq}.\mspace{14mu} 28} \right)\end{matrix}$

Consider the following partition of the posterior covariance matrix

$\begin{matrix}{{C_{n} = \begin{pmatrix}T_{n} & X_{n} \\X_{n}^{H} & Y_{n}\end{pmatrix}},} & \left( {{Eq}.\mspace{14mu} 29} \right)\end{matrix}$

where T_(n) is the M_(T)×M_(T) marginal covariance matrix (M_(T)<M).Consider also the following partition of the sensitivity kernel g_(n+1)

$\begin{matrix}{{g_{n + 1} = \begin{pmatrix}{\overset{\_}{g}}_{n + 1} \\h_{n + 1}\end{pmatrix}},} & \left( {{Eq}.\mspace{14mu} 30} \right)\end{matrix}$

where g _(n+1) is the “restricted” sensitivity kernel. It is restrictedin the sense that it has just M_(T) components. In Eq. 30, h_(n+1)represents the remainder of the sensitivity kernel g_(n+1) that isoutside the particular region of interest. The restricted kernel g_(n+1) is the part of the sensitivity kernel that is just sensitive tothe model parameters in the particular region of interest. Therefore,the updated marginal covariance matrix is given by

$\begin{matrix}\begin{matrix}{T_{n + 1} = {T_{n} - \frac{{\sigma_{n + 1}^{- 2}\left( {{T_{n}{\overset{\_}{g}}_{n + 1}^{*}} + {X_{n}h_{n + 1}^{*}}} \right)}\left( {{{\overset{\_}{g}}_{n + 1}^{T}T_{n}} + {h_{n + 1}^{T}X_{n}^{H}}} \right)}{1 + {\sigma_{n + 1}^{- 2}g_{n + 1}^{T}C_{n}g_{n + 1}^{*}}}}} \\{{\equiv {T_{n} - {a_{n + 1}a_{n + 1}^{H}} - {a_{n + 1}b_{n + 1}^{H}} - {b_{n + 1}a_{n + 1}^{H}} - {b_{n + 1}b_{n + 1}^{H}}}},}\end{matrix} & \begin{matrix}{\left( {{Eq}.\mspace{14mu} 31} \right)\;} \\\; \\\left( {{Eq}.\mspace{14mu} 32} \right)\end{matrix}\end{matrix}$

where the vectors a_(n+1) and b_(n+1) are defined as follows

$\begin{matrix}{{a_{n + 1} \equiv \frac{\sigma_{n + 1}^{- 1}T_{n}{\overset{\_}{g}}_{n + 1}^{*}}{\sqrt{1 + {\sigma_{n + 1}^{- 2}g_{n + 1}^{T}C_{n}g_{n + 1}^{*}}}}},} & \left( {{Eq}.\mspace{14mu} 33} \right) \\{{b_{n + 1} \equiv \frac{\sigma_{n + 1}^{- 1}X_{n}h_{n + 1}^{*}}{\sqrt{1 + {\sigma_{n + 1}^{- 2}g_{n + 1}^{T}C_{n}g_{n + 1}^{*}}}}},} & \left( {{Eq}.\mspace{14mu} 34} \right)\end{matrix}$

Note that it is unnecessary to expand the denominator in terms of thepartitions T_(n), X_(n), g _(n+1), and h_(n+1). Eq. 32 shows that theupdated marginal covariance matrix, T_(n+1), is a rank-four update ofT_(n). Recursive application of the matrix determinant lemma yields

                                        (Eq.  35)$\frac{\det \left( T_{n} \right)}{\det \left( T_{n + 1} \right)} = \left\{ {{\left\lbrack {1 - {a_{n + 1}^{H}T_{n}^{- 1}a_{n + 1}}} \right\rbrack \left\lbrack {1 - {b_{n + 1}^{H}\left( {T_{n} - {a_{n + 1}a_{n + 1}^{H}}} \right)^{- 1}a_{n + 1}}} \right\rbrack} \times {\quad{\quad{\left\lbrack {1 - {{a_{n + 1}^{H}\left( {T_{n} - {a_{n + 1}a_{n + 1}^{H}} - {a_{n + 1}b_{n + 1}^{H}}} \right)}^{- 1}b_{n + 1}}} \right\rbrack \times \mspace{695mu} \left( {{Eq}.\mspace{14mu} 36} \right)\mspace{175mu} {\left. \quad\left\lbrack {1 - {{b_{n + 1}^{H}\left( {T_{n} - {a_{n + 1}a_{n + 1}^{H}} - {a_{n + 1}b_{n + 1}^{H}} - {b_{n + 1}a_{n + 1}^{H}}} \right)}^{- 1}b_{n + 1}}} \right\rbrack \right\}^{- 1}.}}}}} \right.$

The Sherman-Morrison formula gives the rank-one update of the inverse ofa matrix and it may be used to expand the right-hand side of Eq. 35. Thefinal result would be an expression that involves just one matrixinverse, namely T_(n) ⁻¹, and this through inner products with vectors.The latter can be efficiently evaluated using a Cholesky decompositionof T_(n).

Stacking

It is desirable to achieve a better signal-to-noise ratio in measurementdata acquired by a survey arrangement. Poor signal-to-noise ratio canresult in reduced resolution of model parameters.

To improve signal-to-noise ratio in a survey design, techniques ormechanisms according to some implementations can implement stacking, inwhich an observation associated with a particular survey setting can beselected multiple times. For example, a particular source-receiver pair(a combination of a seismic source and a seismic sensor) in a surveyarrangement can be selected multiple times to collect observations atdifferent times. The observations of this particular source-receiverpair can then be combined (a process referred to as “stacking”) toimprove the signal-to-noise ratio of the measurement data acquired bythe particular source-receiver pair.

Selecting a particular observation multiple times can refer to using thesame source-receiver pair multiple times, as noted above. Alternatively,selecting a particular observation multiple times can refer to selectinga first source-receiver pair, as well as at least a secondsource-receiver pair (e.g., same source but another seismic sensor thatis nearby, to within some predefined distance, to the seismic sensor ofthe first source-receiver pair).

In the context of performing survey design as discussed above, selectingan observation multiple times refers to selecting a particular entry ofa sensitivity matrix (or selecting a particular sensitivity datastructure) multiple times, which can result in a reduction in the modeluncertainty.

The following discusses an example in which the same observation isselected twice in a survey system that has a particular source-receiverpair and two scatterers placed at distances 1 m and (1+Δ) m from thesource-receiver pair. A “scatterer” refers to an element in the targetstructure that causes scattering of a wavefield (e.g., seismicwavefield). In the context of a subterranean structure, a scatter canrepresent an interface of the subterranean structure betweeninhomogeneous structures that cause scattering of wavefields.

The foregoing example survey system is further assumed to include justtwo potential observations: the observation for which the wavepropagation constant is k₁ and the observation for which the wavepropagation constant is k₂>k₁. The wave propagation constant representsa characteristic of the target structure that affects propagation ofwavefields (e.g., seismic wavefields).

The observation with wave propagation constant is k₁ is referred to asthe first observation, and the observation with wave propagationconstant is k₂ is referred to as the second observation. Let the modelprior covariance matrix and the data prior covariance matrix be,respectively:

$\begin{matrix}{{C = {\sigma^{2}\begin{bmatrix}1 & ^{- \frac{\Delta}{L}} \\^{- \frac{\Delta}{L}} & 1\end{bmatrix}}},{C_{D} = {\begin{bmatrix}\sigma_{1}^{2} & 0 \\0 & \sigma_{2}^{2}\end{bmatrix}.}}} & \left( {{Eq}.\mspace{14mu} 67} \right)\end{matrix}$

According to Eq. 67, the data is assumed to be uncorrelated and themodel parameters are assumed to be correlated with correlation length L.Upon using the one-dimensional Green kernels, the sensitivity matrix forthis system takes on the form

$\begin{matrix}{G = {\frac{1}{4}\begin{bmatrix}^{2\; k_{1}} & ^{2{{({1 + \Delta})}}\; k_{1}} \\^{2\; k_{2}} & ^{2{{({1 + \Delta})}}\; k_{2}}\end{bmatrix}}} & \left( {{Eq}.\mspace{14mu} 68} \right)\end{matrix}$

Consequently, the C-norms corresponding to the first and secondobservations are, respectively,

$\begin{matrix}{{\frac{\sigma^{2}}{8\sigma_{1}^{2}}\left\lbrack {1 + {^{- \frac{\Delta}{L}}{\cos \left( {2\Delta \; k_{1}} \right)}}} \right\rbrack},{{\frac{\sigma^{2}}{8\sigma_{1}^{2}}\left\lbrack {1 + {^{- \frac{\Delta}{L}}{\cos \left( {2\Delta \; k_{2}} \right)}}} \right\rbrack}.}} & \left( {{Eq}.\mspace{14mu} 69} \right)\end{matrix}$

If an algorithm is to favor the first observation over the secondobservation, i.e., if the algorithm is to suggest the counter-intuitivesituation where the lower-frequency observation is favored over thehigher-frequency observation, the data noises and the model varianceswould have to satisfy

$\begin{matrix}{\frac{\sigma_{2}^{2}}{\sigma_{1}^{2}} > {\frac{1 + {^{- \frac{\Delta}{L}}{\cos \left( {2\Delta \; k_{2}} \right)}}}{1 + {^{- \frac{\Delta}{L}}{\cos \left( {2\Delta \; k_{1}} \right)}}}.}} & \left( {{Eq}.\mspace{14mu} 70} \right)\end{matrix}$

This condition is an intricate relationship between the model and datauncertainties, the correlation length, the wavelengths (or frequencies)involved, and the separation between scatterers.

The foregoing assumes the circumstance under which the algorithm selectsthe first observation. After updating the posterior model covariance,the following new C-norms are obtained:

$\begin{matrix}{\frac{\sigma^{2}\left( {1 + {^{- \frac{\Delta}{L}}{\cos \left\lbrack {2\Delta \; k_{1}} \right\rbrack}}} \right)}{{\sigma^{2}\left( {1 + {^{- \frac{\Delta}{L}}{\cos \left\lbrack {2\Delta \; k_{1}} \right\rbrack}}} \right)} + {8\sigma_{1}^{2}}},\frac{\begin{matrix}{{{\sigma^{4}\left( {^{- \frac{2\Delta}{L}} - 1} \right)}{\cos^{2}\left\lbrack {2{\Delta \left( {k_{1} - k_{2}} \right)}} \right\rbrack}} +} \\{8\sigma_{1}^{2}{\sigma^{2}\left\lbrack {1 + {^{- \frac{\Delta}{L}}{\cos \left( {2\Delta \; k_{2}} \right)}}} \right\rbrack}}\end{matrix}}{8{\sigma_{2}^{2}\left\lbrack {{\sigma^{2}^{- \frac{\Delta}{L}}{\cos \left( {2\Delta \; k_{1}} \right)}} + \sigma^{2} + {8\sigma_{1}^{2}}} \right\rbrack}},} & \left( {{Eq}.\mspace{14mu} 71} \right)\end{matrix}$

where the expression on the left corresponds to the first observationand the expression on the right corresponds to the second observation.Setting the left-hand-side expression larger than the right-hand-sideexpression can yield improved resolution of the model. If the foregoingcondition is satisfied, then stacking of measurements (of a selectedobservation) may improve the resolution of the model.

Determining Survey Design for a Model Having a Dip Scatterer

In some cases, inhomogeneities within a subterranean structure to becharacterized can be modeled as a discrete set of point scatterers. Theelastic radiation (corresponding to scattered seismic wavefields)emanating from each point scatterer can be assumed to be essentiallyisotropic (the scattering radiation is invariant with direction).However, in practice, the stratified structure of the subterraneanstructure often gives rise to inhomogeneities for which scatteringradiation is generally anisotropic. As a result, modelinginhomogeneities as point scatterers with isotropic characteristics maynot produce accurate results.

To address the foregoing, techniques or mechanisms according toalternative implementations can include, in a model of a targetstructure, information relating to scatterers (which correspond toinhomogeneities in the target structure) with having anisotropicradiation patterns. Such scatterers can be referred to as dippingscatterers, since each such scatterer can be considered to represent adipping interface in the target structure that has an anisotropicradiation pattern when scattering wavefields. As discussed in furtherdetail, dipping scatterers can be represented as a scatterer having afinite size (rather than just a point).

For uncorrelated data noise, Eq. 13 (discussed further above) reduces tothe following iterative expression:

$\begin{matrix}{{\xi_{opt}^{(n)} = {\arg \mspace{11mu} {\max\limits_{\xi \in \Xi}{\gamma_{n + 1}}_{C_{n}^{*}}^{2}}}},{{for}\mspace{14mu} a\mspace{14mu} {fixed}\mspace{14mu} {\Xi }}} & \left( {{Eq}.\mspace{14mu} 72} \right)\end{matrix}$

where γ_(n+1)≡σ_(n+1) ⁻¹g_(n+1), σ_(n+1) is the data standard deviationand g_(n+1) ^(T) is the sensitivity kernel of a candidate observation(i.e., an entry, such as a row, of the sensitivity matrix). The C*-normis defined as ∥γ_(n+1)∥_(C) _(n) ₊ ²≡γ_(n+1) ^(H)C*_(n)γ_(n+1), where Hstands for the Hermitian conjugate and * for the complex conjugate.

In the frequency domain, the general equation governing elastodynamicphenomena can be written as

ω²ρ(x)u _(i)(x,ω)+Σ_(j,k,l)∂_(j) [c _(ijkl)(X)∂_(k) U ₁(x,ω)]=s_(i)(x,ω)  (Eq. 73)

where u_(i)(x,ω) is the i^(th) component of displacement at point xεVand frequency ω, ∂_(j)≡∂/∂x_(j), and s_(i)(x,ω) the source term. Thissource term is defined in terms of the volume density of forcef_(i)(x,ω) and the volume density of moment M_(ij)(x,ω) as

s _(i)(x,ω)≡f _(i)(x,ω)+Σ_(j)∂_(j) M _(ij)(x,ω).  (Eq. 74)

The solution of Eq. 73 may be expressed as a function of the outgoingGreen function

(x, x′, ω) as

u _(i)(x,ω)=−∫_(V)Σ_(j)

(x,x′,ω)s _(j)(x′,ω)d ³ x′.  (Eq. 75)

If it is assumed that the propagation medium (target subterraneanstructure) represented by the vector m(x)≡(ρ(x), {c_(ijkl)(x)}) isviewed as the superposition of a reference medium represented by thevector m⁽⁰⁾(x)≡(ρ⁽⁰⁾(x), {c_(ijkl)(x))} and a perturbation modelrepresented by the vector Δm (x)≡(Δρ(x), {Δc_(ijkl)(x)}) so that

ρ(x)=ρ⁽⁰⁾(x)+Δρ(x)

c _(ijkl)(x)=c _(ijkl) ⁽⁰⁾(x)Δc _(ijkl)(x).  (Eq. 76)

Note that Eq. 76 is similar to Eq. 3 set forth further above. Under theBorn approximation, the scattered displacement fields can be written as

Δu _(i)(x,ω)=ω²∫_(V)Σ_(i)

(x,x′,ω)u _(j)Δρ(x′,ω)Δρ(x′)d ³ x′

−∫_(V)Σ_(j,k,l,n)∂_(k)[

(x,x′,ω)]Δc _(nklj)(x′,ω)d ³ x′,  (Eq. 77)

wherein

(x, x′, ω) is the outgoing Green function in the unperturbed medium andu_(i) ⁽⁰⁾(x′, ω) is the unperturbed field in the reference medium givenby

(x,ω)=−∫_(V)Σ_(j)

(x,x′,ω)s _(j)(x′,ω)d ³ x′.  (Eq. 78)

In Eq. 77, there is no assumption regarding any particular source typenor is there any assumption regarding any particular structure for theinhomogeneities in the propagation medium. To account for anisotropicscattering radiation in the propagation medium, inhomogeneities in thepropagation medium can be represented as one-dimensional ortwo-dimensional scattering elements (the dipping scatterers discussedfurther above).

It may be sufficient to assume that the contrasts in elastic parametersare constant over the extent of each individual scattering element,although the elastic parameters may vary from one scattering element toanother. Thus, the model parameters (density and stiffness coefficientperturbations) corresponding to M scattering elements with respectiveconstant contrasts Δρ_(β) and (Δc_(ijkl))_(β), β=1, 2, . . . , M, aregiven by

Δρ(x)=Σ_(β=1) ^(M)Δρ_(β)Ψ_(β) ^((D))(x)

Δc _(ijkl)(x)=Σ_(β=1) ^((6×6)M)(Δc _(ijkl))_(β)Ψ_(β) ^((D))(x)  (Eq. 79)

wherein the Ψ_(β) ^((D))'s are indicator functions that encode the shapeof the scattering elements.) Thus, information pertaining to dippingscatterers are included in models by use of the Ψ_(β) ^((D)) functions.

According to Huygens's principle, a D-dimensional scatterer

, D=1, 2, . . . , may be viewed as including a continuous set of pointscatterers. Thus, the form factors may be defined as

Ψ_(β) ^((D))(x)≡

δ(x−y _(β)(Λ^((D))))dΛ ^((D)),  (Eq. 80)

where δ(•) stands for the Dirac delta and Λ^((D)) stands for the set ofintegration parameters that characterize the spatial extent of thescattering elements. For a line-segment

, Λ⁽¹⁾ is a real number. However, when the scattering elements aretwo-dimensional elements (e.g., two-dimensional disks),

and Λ⁽²⁾ take on the form of a pair of real numbers: the first realnumber spans the radial extent of the two-dimensional element and thesecond number represents the element's sweeping angle (i.e., Λ⁽²⁾≡(ν,φ)≡[0,1]×[0,2π] for instance). Substituting Eq. 80 into Eq. 77 andtaking into account the definition of Ψ_(β) ^((D))(x) yields

Δ   u i  ( x , ω ) = ∑ β = 1 M  ω 2  Δρ β   ∑ j  ij ( 0 )  ( x, y β  ( Λ ( D ) ) , ω )  u j ( 0 )  ( y β  ( Λ ( D ) ) , ω )   Λ( D ) - ∑ β = 1 M   ∑ j , k , l , n  ( Δ   c nklj ) β  [ ∂ k ′  i  n ( 0 )  ( x , x ′ , ω )  ∂ l ′  u j ( 0 )  ( x ′ , ω ) ] x ′ =y β  ( Λ ( D ) )   Λ ( D ) . ( Eq .  81 )

Eq. 81 may be written concisely as

Δu _(i)≡Σ_(β=1) ^((1+6×6)M) G _(iβ) Δm _(β),  (Eq. 82)

where {Δm_(β)}_(1≦β≦M) represent the mass density contrasts,{Δm_(β)}_(M+1≦β≦(6×6)M) represent the stiffness contrasts, and G_(iβ)are the entries of a possibly complex N×(1+6×6)M sensitivity matrix. Nand (1+6×6)M are the number of displacement observations and the numberof model parameters, respectively. The entries of the sensitivitymatrix, G_(iβ), can be split into two categories: entries G_(iβ) ^((p))that represent the contributions associated with the mass densitycontrasts Δp_(β), and G_(iβ) ^((c)) that represent the contributionsassociated with the stiffness contrasts (Δc_(nklj))_(β). They are givenby

G i   β ( ρ ) ≡ ω 2   ∑ j  ij ( 0 )  ( x , y β  ( Λ ( D ) ) , ω ) u j ( 0 )  ( y β  ( Λ ( D ) ) , ω )   Λ ( D )   G i   β ( c )≡ -  [ ∂ k ′  i   n ( 0 )  ( x , x ′ , ω )  ∂ l ′  u j ( 0 )  (x ′ , ω ) ] x ′ = y β  ( Λ ( D ) )   Λ ( D ) . ( Eq .  83 )

The introduction of a spatial extent for the scattering elementsmanifests itself as an overall factor that multiplies the matrix entriesof the point-scatterers case. This factor reduces to unity when thespatial extent vanishes and reproduces the effects of specularreflection (point scattering). The spatial extent of the scatteringelement can be used as a controllable parameter to characterize theuncertainty on a structural dipping interface—a higher confidence in thedipping character of the scattering structure may translate into largersizes for the scattering element, thereby making the scattering morespecular. For instance, in the 2D acoustic example, a heuristic rule ofthumb could be

l:φ ⁻¹:σ_(dip) ⁻¹,  (Eq. 84)

where l is the length of the line-segment scatterer, is a measure of thebeam width which could be, for instance, the half-power beamwidth (HPBW)or the first-null beamwidth (FNBW) Balanis-bk, and σ_(dip) is thedip-angle standard deviation. The parameter φ is identified withσ_(dip). Introducing the normalized length 0≦ l≡l/λ<∞ and the normalizedstandard deviation 0≦σ_(dip)≡σ_(dip)/π≦1, one can devise another simpleexpression that is more quantitatively meaningful than Eq. 84 and whichhas the appropriate asymptotic behavior, namely

$\begin{matrix}{\overset{\_}{l} \equiv {\frac{1}{{\overset{\_}{\sigma}}_{dip}} - 1.}} & \left( {{Eq}.\mspace{14mu} 85} \right)\end{matrix}$

Given a particular uncertainty on the dip, σ _(dip), Eq. 85 allows forthe calculation of the size (represented as l) of the scatteringelement.

Computing Environment

FIG. 5 is a block diagram of a computing system 500 that includes aprocessing module 502 that is able to perform tasks of FIGS. 3 and 4.The processing module 502 is executable on one or multiple processors504. A processor can include a microprocessor, microcontroller,processor module or subsystem, programmable integrated circuit,programmable gate array, or another control or computing device. Theprocessing module 502 can be implemented as machine-readableinstructions.

The computing system 500 includes a storage medium 506 (or storagemedia). In addition, the computing system 500 includes a networkinterface 508 that can communicate over a network. The storage medium(or storage media) 506 can be used to store sensitivity data structures510, as well as other data structures discussed above.

The storage medium 506 (or storage media) can be implemented as one ormore computer-readable or machine-readable storage media. The storagemedia include different forms of memory including semiconductor memorydevices such as dynamic or static random access memories (DRAMs orSRAMs), erasable and programmable read-only memories (EPROMs),electrically erasable and programmable read-only memories (EEPROMs) andflash memories; magnetic disks such as fixed, floppy and removabledisks; other magnetic media including tape; optical media such ascompact disks (CDs) or digital video disks (DVDs); or other types ofstorage devices. Note that the instructions discussed above can beprovided on one computer-readable or machine-readable storage medium, oralternatively, can be provided on multiple computer-readable ormachine-readable storage media distributed in a large system havingpossibly plural nodes. Such computer-readable or machine-readablestorage medium or media is (are) considered to be part of an article (orarticle of manufacture). An article or article of manufacture can referto any manufactured single component or multiple components. The storagemedium or media can be located either in the machine running themachine-readable instructions, or located at a remote site from whichmachine-readable instructions can be downloaded over a network forexecution.

In the foregoing description, numerous details are set forth to providean understanding of the subject disclosed herein. However,implementations may be practiced without some of these details. Otherimplementations may include modifications and variations from thedetails discussed above. It is intended that the appended claims coversuch modifications and variations.

1. A method comprising: calculating complex-valued sensitivity datastructures corresponding to respective candidate survey settings,wherein the sensitivity data structures relate measurement dataassociated with a target structure to at least one parameter of a modelof the target structure; and selecting, based on the sensitivity datastructures, a subset of the candidate survey settings according to acriterion for enhancing resolution in characterizing the targetstructure.
 2. The method of claim 1, wherein the criterion is based onidentifying at least one of the candidate survey settings that reducesposterior uncertainty.
 3. The method of claim 1, wherein selecting thesubset comprises performing an iterative procedure, the iterativeprocedure comprising: adding one of the sensitivity data structures to acollection of sensitivity data structures previously considered; andbased on a present content of the collection of the sensitivity datastructures, identifying, according to the criterion, one of thecandidate survey settings corresponding to the present content of thecollection.
 4. The method of claim 3, wherein the iterative procedurefurther comprises: determining whether a convergence condition has beensatisfied; and if the convergence condition has not been satisfied,continuing with the iterative procedure by adding a further sensitivitydata structure to the collection, and based on a further present contentof the collection, identifying, according to the criterion, one of thecandidate survey settings corresponding to the further present contentof the collection.
 5. The method of claim 1, further comprising:configuring survey equipment according to the at least one surveysetting to perform a survey operation of the target structure.
 6. Themethod of claim 1, wherein a particular one of the candidate surveysettings includes at least one item selected from the group consistingof: type of survey equipment, a position of at least one component ofthe survey equipment, at least one operational characteristic of thesurvey equipment.
 7. The method of claim 1, wherein the criterion isaccording to a ratio between a determinant of a prior covariance matrixand a determinant of a posterior covariance matrix.
 8. The method ofclaim 1, wherein the sensitivity data structures include informationrelating to the at least one model parameter that is sensitive to asubregion less than an entirety of the target structure.
 9. The methodof claim 1, wherein the at least one parameter of the model accounts foran anisotropic radiation pattern of a scatterer in the target structure.10. The method of claim 9, wherein the at least one parameter of themodel is based on a function that encodes a shape of the scatterer. 11.The method of claim 1, wherein selecting the subset of the candidatesurvey settings comprises selecting a particular candidate surveysettings plural times to allow for stacking of measurement data acquiredby the particular candidate survey setting.
 12. The method of claim 1,further comprising using a process including the calculating andselecting to identify, for further processing, a subset of measurementdata acquired by a survey arrangement.
 13. An article comprising atleast one machine-readable storage medium storing instructions that uponexecution cause a system to: calculate complex-valued sensitivity datastructures corresponding to respective candidate survey settings,wherein the sensitivity data structures relate measurement dataassociated with a target structure to at least one parameter of a modelof the target structure; and select, based on the sensitivity datastructures, a subset of the candidate survey settings according to acriterion for enhancing resolution in characterizing the targetstructure.
 14. The article of claim 13, wherein the sensitivity datastructures relate perturbations of measurement data to perturbations ofthe at least one model parameter.
 15. The article of claim 13, whereinthe target structure comprises a target subterranean structure.
 16. Thearticle of claim 13, wherein the criterion is based on prior covarianceinformation relating to the model.
 17. The article of claim 16, whereinthe criterion is based on a value derived from a determinant of theprior covariance information and a determinant of posterior covarianceinformation.
 18. The article of claim 13, wherein the selecting isperformed using an iterative procedure in which individual ones of thesensitivity data structures are added to a collection of sensitivitydata structures for consideration with successive iterations.
 19. Thearticle of claim 13, wherein the at least one parameter of the modelaccounts for an anisotropic radiation pattern of a scatterer in thetarget structure.
 20. A system comprising: at least one storage mediumto store complex-valued sensitivity data structures corresponding torespective candidate survey settings, wherein the sensitivity datastructures relate measurement data associated with a target structure toat least one parameter of a model of the target structure; and at leastone processor to select, based on the sensitivity data structures, asubset of the candidate survey settings according to a criterion forenhancing resolution in characterizing the target structure.
 21. Thesystem of claim 20, wherein the criterion is based on prior covarianceinformation relating to the model.